A Friendly Question about God and Logic

Here’s a thoughtful email query I received with the title “Friendly Question about God and Logic”:

Recently, I have been reading about God and abstract objects and came across your article in Phil-Christi with Greg Welty regarding God and logic. I thoroughly enjoyed it and found it both persuasive and useful. In doing further reading on your website I came across a follow up article where you argue that atheism presupposes theism (and so does every other ism) and your argument gets close to an objection to the claim that logic depends on God that I have long wondered about. In the article, in reference to Atheism you argue the following:

(1) God does not exist. [assumption for reductio]
(2) It is true that God does not exist. [from (1)]
(3) There is at least one truth (namely, the truth that there is no God). [from (2)]
(4) If there are truths, they are divine thoughts.
(5) There is at least one divine thought. [from (3) and (4)]
(6) If there are divine thoughts, then God exists.
(7) Therefore, God exists. [from (5) and (6)]

Consider the following reconstruction:

(1*) God does not exist. [assumption for reductio]
(2*) It is true that God does not exist. [from (1*)]
(3*) There is at least one truth (namely, the truth that there is no God). [from (2*)]
(4*) Therefore, truth does not depend on God. [from (1*) and (3*)]

Let me make explicit why I think (1*-4*):

(5*) The laws of logic are divine thoughts.
(6*) According to the aseity-sovereignty doctrine if God did not exist then nothing would exist.
(7*) If God did not exist there would be no divine thoughts.
(8*) Therefore, there would be no laws of logic.

But if (5*-8*) hold, the proposition either God exists or He does not, would be a truthful description of that state of affairs and be an instance of the LEM. Likewise, the proposition God exists, would be false, not true. Not both true and false, thus an instance of the LNC.

Or another way of stating it would be:

If God did not exist then nothing would exist. But it seems that even if God did not exist there would be at least one thing that would exist, the state of affairs, nothing exists. Doesn’t that imply/entail that there is at least one truth about that state of affairs, the truth nothing exists? If that is the case don’t we have laws of logic?

I assume my objection is misguided in some way. If you have time to address this question and clarify my error I would appreciate it.

I think there are a few problems with the way the objection has been posed. For example, surely it’s incoherent to claim that if nothing were to exist, at least one thing would exist: the state of affairs, nothing exists. (What we have here, in effect, is a reductio ad absurdum of the claim that possibly nothing exists. A more rigorous argument, courtesy of the Maverick Philosopher, can be found here.) Moreover, I wouldn’t consider the proposition either God exists or He does not to be a description of a state of affairs, but rather a necessary logical truth.

Quibbles aside, however, the thrust of the objection is clear enough. I would restate it this way:

(1) If God did not exist, there would be at least one truth: the truth that God does not exist.
(2) If there are truths, there are instances of the laws of logic.
(3) If there are instances of the laws of logic, there are laws of logic.
(4) Therefore, if God did not exist, there would be laws of logic. [from (1), (2), and (3)]
(5) Therefore, the laws of logic can exist independently of God. [from (4)]
(6) Therefore, the laws of logic are not divine thoughts. [from (5)]

Assuming that this is a fair restatement of the objection, where does it go wrong? I think the inferences from (4) to (5) and from (5) to (6) are valid. I suppose premises (2) and (3) could be challenged, but I’m happy to grant them for the sake of argument. The real problem lies with (1) insofar as it begs the question against my position. Greg Welty and I gave a lengthy argument for the claim that propositions are divine thoughts. If our argument is sound then the following is true:

(1′) If God did not exist, there would be no truths.

Clearly (1) and (1′) cannot both be true; if (1) is true then (1′) is false. Hence any objection that relies on (1) actually assumes what it needs to prove, namely, that our argument is unsound.

I suppose one might reply that (1) is self-evidently true, or at least follows from a self-evident truth. Consider this generalization:

(*) If X did not exist, there would be at least one truth: the truth that X does not exist.

Isn’t this a self-evident principle? At first glance, one might think so. But then (1) follows trivially as an instance of (*).

However, this is too hasty, because the following is also an instance of (*):

(7) If truths did not exist, there would be at least one truth: the truth that truths do not exist.

Surely (7) is incoherent and therefore necessarily false. (This suggests, of course, an argument for the necessary existence of truths.) At any rate, we have good reason to think that (*) cannot be wielded in support of (1).

The fact is that (1) isn’t a trivial or obvious truth at all. It’s a counterpossible conditional (assuming that God exists necessarily if God exists at all) and there’s serious philosophical debate over how to interpret and evaluate such claims. (Compare the following claim: If 3 were equal to 2, 9 would be an even number. Is that claim true or false? Hard to say!) At a minimum, whether or not you think (1) is true will depend, in part, on what you already believe about the relationship between God and propositions (and likewise for the relationship between God and logic).

All this to say, it seems to me that objections like the one expressed above beg the question against the argument for God from logic, since they rely on counterpossible conditionals which tacitly presuppose that propositions are metaphysically independent of God.

(where (*) is a premise for reductio). The Friendly Objector also infers (**) from (*) on the way to a rather different conclusion.

You object to the latter argument on the grounds (perhaps among other things) that it depends on the counterpossible:

(1) If God did not exist, there would be at least one truth

whereas on your (or your and Welty’s) view:

(1′) If God did not exist, there would be no truths

And you add: “Clearly (1) and (1′) cannot both be true; if (1) is true then (1′) is false. Hence any objection that relies on (1) actually assumes what it needs to prove, namely, that our argument is unsound.” Further: “The fact is that (1) isn’t a trivial or obvious truth at all. It’s a counterpossible conditional … and there’s serious philosophical debate over how to interpret and evaluate such claims. (Compare the following claim: If 3 were equal to 2, 9 would be an even number. Is that claim true or false? Hard to say!).”

And here is where my confusion creeps in. It is double. First, if on your view (1′) but not (1) is true, then in virtue of what exactly may we infer (2) on the strength of (1) in a way that supports your argument but not that of the Friendly Objector? If there would have been no truths if God had not existed, then how would it have been true that God doesn’t exist in that case? Aren’t both inferences in the same boat?

The second confusion is like unto the first. Both (1) and (1′) are counterpossibles, and are therefore (so far forth) equally subject to the vicissitudes of that peculiar breed. But as the initial step in both arguments above apparently depend on (1) — or at all events on:

If God did not exist, it would have been true that God does not exist

— doesn’t the controversial nature of counterpossibles cast doubt on both arguments? (And, as a separate matter, doesn’t it infect (1′)? Further, if, as on the “standard” counterfactual semantics, counterpossibles are all vacuously true, it’s just false that (1) and (1′) can’t both be true. Anyway, there are indeed interesting broadly semantical issues in drawing distinctions among relevant pairs of counterpossibles, and this is a matter worthy of attention. As is the fact that philosophers do appear to rely on distinctions among counterpossibles in fairly uncontroversial ways.)

Thanks for the comments, Paul! (Thanks also to Ben for his magnanimous remarks in my defense.)

Let me address your ‘confusions’ in reverse order.

1. My point about the debate over counterpossibles (misleadingly stated, perhaps) was not to cast doubt on all use of counterpossibles. It was only to observe that we shouldn’t think (1) is obviously or trivially true.

Yes, on the “standard” counterfactual semantics, counterpossibles are all vacuously true. But that’s one reason I don’t accept the “standard” semantics. It makes no sense to me (and I daresay to many other people) that (1) and (1′) could both be true. If certain counterfactual semantics have that consequence, so much the worse for those semantics. Similarly, it makes no sense to me that propositions like If there were no God, there would be divine thoughts or If there were no propositions, there would be truths should be considered true.

2. Now, to your charge that my original argument depends on the same counterpossible as the FO’s. Like Ben, I don’t think that the inference from God does not exist to It is true that God does not exist needs to be licensed by a counterpossible (i.e., If God did not exist, it would have been true that God does not exist). It could be licensed something as simple as If P then it is true that P. At any rate, I think most people would be willing to grant that the second claim follows logically from the first without disputing over exactly what licenses that deduction.

Furthermore, my original argument doesn’t really need that initial inference; it could begin just as well with the second claim. The argument, as I’m sure you recognize, is directed at affirmations and denials (atheism, agnosticism, and other “isms”). Insofar as any affirmation (or denial) presupposes the existence of at least one truth, the substance of the argument stands. So, for example, if the atheist affirms It is true that God does not exist then we’re up and running.

You rightly observe that FO’s initial argument begins with the same first premise and inference as my original argument. However, my restatement of the objection (which is based more on his “another way of stating it”) was designed to get at the heart of the objection, which I do think depends on a question-begging counterpossible. The crux of my rebuttal is this: the objection tries to prove that the laws of logic (or truths in general) are metaphysically independent of God, but it does so by relying on that question-begging counterpossible.

“(1) God does not exist. [assumption for reductio]
“(2) It is true that God does not exist. [from (1)]
“(3) There is at least one truth (namely, the truth that there is no God). [from (2)]”

Our Friendly Objector, appealing to just that argument, offers another that also begins:

“(1*) God does not exist. [assumption for reductio]
“(2*) It is true that God does not exist. [from (1*)]
“(3*) There is at least one truth (namely, the truth that there is no God). [from (2*)]”

While the arguments subsequently diverge (and the Friendly Objector’s argument is no reductio), the inferential steps, so far forth, are identical. (Are the principles on the basis of which the inferences are drawn the same. Hard to say, as neither you nor the Friendly Objector show your work.)

Now, you reconstruct the initial segment of the Friendly Objector’s argument as:

“(1) If God did not exist, there would be at least one truth: the truth that God does not exist”

and then object:

“The real problem lies with (1) insofar as it begs the question against my position. Greg Welty and I gave a lengthy argument for the claim that propositions are divine thoughts. If our argument is sound then the following is true:

(1′) If God did not exist, there would be no truths.”

The objection here, presumably, is that armed with (1′), the Friendly Objector is not entitled to (3*) on the basis of the steps preceding it.

And here is my comment: If that’s the case, then the same objection tanks the inference to (3) in that (so far) entirely parallel argument you previously offered.

Of course an argument (as Feyerabend taught us) is not a confession, and perhaps with that argument you are offering an ad hominem argument (broadly construed) directed to someone who is not already committed to (1′). Perhaps. In that case there is more to say. But at all events I wanted to try again to make this very modest point.

A couple of other quick notes. You write:

“I don’t think that the inference from God does not exist to It is true that God does not exist needs to be licensed by a counterpossible (i.e., If God did not exist, it would have been true that God does not exist). It could be licensed something as simple as If P then it is true that P. At any rate, I think most people would be willing to grant that the second claim follows logically from the first without disputing over exactly what licenses that deduction.”

Well, given your reconstruction of the Friendly Objector’s argument, I should have highlighted the the initial three steps of the arguments, not the first two. That was a mistake on my part. My comment is (and was) meant to apply to the segments (1)-(3), (1*)-(3*).

I should add that I’m generally a fan of that pattern of inference; on many occasions I have argued that way myself. (Suppose there are truths cold have been false….) I do, however, think there are at least initially plausible reasons for rejecting the validity of that pattern.

Anyway, do these steps, depend on the truth of a counterpossible? Perhaps; perhaps not. Certainly I think that each of (1), (1*) and (2), (2*) is not just false but necessarily false. And so do you. So we are up to our armpits in counterpossibility here. (Of course we can forego explicit counterfactuality altogether if you prefer. An impossibility entails just any proposition you please. Etc.) And, setting the pair:

(~GTr.1) If God did not exist, there would be at least one truth,
(~GTr.2) If God did not exist, there would be no truths

to one side, I should think that of these two:

(~GDT.1) If God did not exist, there would nonetheless have been divine thoughts
(~GDT.2) If God did not exist, there would have been no divine thoughts

the latter enjoys more (perhaps vastly more) warrant for us. To that extent, I think the Friendly Objector just might be on to something. ;)

I had a hard time figuring out what you were trying to say with your first confusion, so if I have misunderstood you please let me know. But as far as I can make out, your concern is as follows: Anderson infers from

(1) God does not exist. [assumption for reductio]

that therefore

(2) It is true that God does not exist. [from (1)]

You ask, why can’t the friendly objector (hereafter “FO”) make the very same move?

The thing is, Anderson isn’t (that I can tell) complaining that FO infers (2) from (1). He’s obviously quite happy to grant that inference. Rather, he’s denying we get to say further that:

FO(1) If God did not exist, there would be at least one truth: the truth that God does not exist.

(I’ve added “FO” to denote the new numbering scheme for FO’s presumed argument.)

You have a second concern, which is that since

FO(1′) If God did not exist, there would be no truths.

is a counterpossible, and we have decided FO(1) is false because it is also counterpossible, then FO(1′) must be false too, contrary to Anderson’s claim that FO(1′) is true. But has Anderson really decided that FO(1) is false because it’s a counterpossible? That’s not how it appears to me. Instead, he decided that FO(1) is false because FO(1′) is true.

You claim this inference from the truth of FO(1′) to the falsity of FO(1) is invalid. Well, at least it’s not obviously valid. Assuming FO(1′) really is a counterpossible (as Anderson is committed to saying), then it’s not clear to me in what sense FO(1′) could be true, or why FO(1) and FO(1′) can’t both be true in whatever sense FO(1′) is true.

So probably Anderson needs to be a little more careful in talking about the truth vs. falsity of FO(1) and FO(1′). But the jist of his complaint seems reasonable to me. I would phrase it this way: Consider the case where FO(1) is a counterpossible; then God exists and we are done. But the only way to consider the case where FO(1) is not a counterpossible is to beg the question against Anderson’s PC argument. So in both cases the objection fails to hold up.

Thanks for your comments, Ben. And my apologies for injecting confusion into these merry Christmas days.

At bottom, I’m suggesting that James is engaged in special pleading in (at least) two ways. First, both arguments share a common piece of counterfactual reasoning. On the assumption that there had been no God, both arguments infer that it would have been true that God doesn’t exist, hence true that something would have been true (i.e., that there would have been some truths). At that point the arguments diverge, and in significant ways. But James complains that the Friendly Objector’s argument (however reconstructed) illicitly appeals to:

(1) If God did not exist, there would be at least one truth

whereas on his and Welty’s view (1) is false and:

(1′) If God did not exist, there would be no truths

is true. But this (I say) is to overlook that his own argument relies (near enough) on (1) as well. The argument James gave is a reductio, where

(*) God does not exist

is the premise for reductio. This assumption is later discharged, but it doesn’t follow that within the scope of the conditional proof just anything goes. Given (*), James infers:

(**) It is true that God doesn’t exist

and from this:

(***) Something is true

We should not be blinded by the English here that this is in fact a piece of counterfactual — and, given that God is a necessary being, counterpossible — reasoning. So, if not (1), then exactly what licenses the inference here? (And if (1′) were the case, pretty clearly neither argument would succeed.)

The second piece of special pleading is this. If there’s “serious philosophical debate over how to interpret and evaluate such claims” as (1), then of course similar strictures apply to (1′). Is the latter claim “claim true or false? Hard to say!”

Well I’m close to agreement with you on the second point. If counterpossibles really are so troublesome (as indeed they appear to be), then I don’t think Anderson should be affirming any of them until we can decide how to treat them. But fortunately for him, none of his arguments rely on affirming counterpossibles. So he can make that correction without doing violence to the rest of his case.

But I still have to disagree with you on the first point. Showing that our logical procedures generate (**) if we start off with (*) is quite different from showing that . The latter seems to say something about what is possible, whereas the former does not. In particular, the latter seems to be saying something like this:

It could be the case that (**), but only if it is also the case that (*).

Without that kind of interpretation, I don’t see how FO’s inferences are warranted. In contrast, Anderson’s reductio doesn’t involve any appeals to what could or could not be the case. Instead we just have an exercise in showing where one sentence, “God does not exist,” logically leads.

By the way, it might interest you to know that Josh Rasmussen recently convinced me that my “truth-in-w” versus “truth-at-w” objection doesn’t work. He has constructed an argument, apparently inspired by Chad Carmichael’s “Universals” (Philosophical Studies 150 [3], 373-89, 2010) which shows that, for any world w and proposition P, []P implies is true at w.

This was very sad for me, because I had written and submitted an article for the secular web library which had already passed peer review, but now I have to withdraw it because Rasmussen undermined it!

I still deny that propositions exist necessarily, but I confess I don’t know how to construct a modal logic which permits me to make this denial. This has led me to profound skepticism concerning the role of modal logic in language. To be honest, it’s rather unsettling.

It’s chapter 4 from that book. I’m not supposed to upload it without permission, but I’m pretty sure he’d be willing to send you a copy if you ask him via email.

As for complete modal skepticism, well, I’m not sure I’m quite there yet. Certain kinds of inferences seem mundane enough, but I will need to watch my step in the problem areas, for instance sliding between “P” and “it is true that P,” or working with propositions whose constituents or referents thereof may not exist in every possible world, etc.

I’ve always been a little bit suspicious of modal inferences. But now I think I have to take that suspicion more seriously than I have in the past.

I will say this though: Even if we grant the necessary existence of propositions and the reliability of ordinary modal logic, then I still have the two other serious problems with your argument. If you recall, I don’t see why thoughts need be essential to the minds to which they belong, if indeed they belong to any mind at all; second, I don’t see why we must identify a necessary conscious being with God. I’ve also been toying with a third potential difficulty, which is that propositions may actually be mereological sums of thoughts instead of individual thoughts.